metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.154D6, C6.302- (1+4), C12⋊Q8⋊37C2, C4⋊C4.210D6, (C4×Dic6)⋊49C2, D6⋊Q8.3C2, C42.C2⋊10S3, Dic3.Q8⋊35C2, (C2×C6).240C24, C42⋊3S3.1C2, D6⋊C4.42C22, C2.59(Q8○D12), C12.3Q8⋊36C2, Dic6⋊C4⋊37C2, (C4×C12).199C22, (C2×C12).602C23, Dic3.13(C4○D4), C4⋊Dic3.316C22, C22.261(S3×C23), Dic3⋊C4.162C22, (C22×S3).105C23, C2.31(Q8.15D6), C3⋊4(C22.35C24), (C2×Dic6).252C22, (C4×Dic3).216C22, (C2×Dic3).260C23, C2.91(S3×C4○D4), C4⋊C4⋊S3.2C2, C6.202(C2×C4○D4), C4⋊C4⋊7S3.13C2, (S3×C2×C4).130C22, (C3×C42.C2)⋊13C2, (C3×C4⋊C4).195C22, (C2×C4).205(C22×S3), SmallGroup(192,1255)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 416 in 192 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2, C3, C4 [×15], C22, C22 [×3], S3, C6 [×3], C2×C4 [×7], C2×C4 [×9], Q8 [×4], C23, Dic3 [×2], Dic3 [×6], C12 [×7], D6 [×3], C2×C6, C42, C42 [×5], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×14], C22×C4, C2×Q8 [×2], Dic6 [×4], C4×S3 [×2], C2×Dic3 [×7], C2×C12 [×7], C22×S3, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C42.C2 [×4], C42⋊2C2 [×4], C4⋊Q8, C4×Dic3 [×5], Dic3⋊C4 [×10], C4⋊Dic3 [×4], D6⋊C4 [×6], C4×C12, C3×C4⋊C4 [×6], C2×Dic6 [×2], S3×C2×C4, C22.35C24, C4×Dic6, C42⋊3S3, Dic6⋊C4, C12⋊Q8, Dic3.Q8 [×3], C12.3Q8, C4⋊C4⋊7S3, D6⋊Q8 [×2], C4⋊C4⋊S3 [×3], C3×C42.C2, C42.154D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2- (1+4) [×2], S3×C23, C22.35C24, Q8.15D6, S3×C4○D4, Q8○D12, C42.154D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 44 7 38)(2 55 8 49)(3 46 9 40)(4 57 10 51)(5 48 11 42)(6 59 12 53)(13 85 19 91)(14 31 20 25)(15 87 21 93)(16 33 22 27)(17 89 23 95)(18 35 24 29)(26 79 32 73)(28 81 34 75)(30 83 36 77)(37 72 43 66)(39 62 45 68)(41 64 47 70)(50 63 56 69)(52 65 58 71)(54 67 60 61)(74 94 80 88)(76 96 82 90)(78 86 84 92)
(1 76 67 18)(2 19 68 77)(3 78 69 20)(4 21 70 79)(5 80 71 22)(6 23 72 81)(7 82 61 24)(8 13 62 83)(9 84 63 14)(10 15 64 73)(11 74 65 16)(12 17 66 75)(25 46 86 50)(26 51 87 47)(27 48 88 52)(28 53 89 37)(29 38 90 54)(30 55 91 39)(31 40 92 56)(32 57 93 41)(33 42 94 58)(34 59 95 43)(35 44 96 60)(36 49 85 45)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 74 19 80)(14 79 20 73)(15 84 21 78)(16 77 22 83)(17 82 23 76)(18 75 24 81)(25 32 31 26)(27 30 33 36)(28 35 34 29)(37 54 43 60)(38 59 44 53)(39 52 45 58)(40 57 46 51)(41 50 47 56)(42 55 48 49)(61 66 67 72)(62 71 68 65)(63 64 69 70)(85 88 91 94)(86 93 92 87)(89 96 95 90)
G:=sub<Sym(96)| (1,44,7,38)(2,55,8,49)(3,46,9,40)(4,57,10,51)(5,48,11,42)(6,59,12,53)(13,85,19,91)(14,31,20,25)(15,87,21,93)(16,33,22,27)(17,89,23,95)(18,35,24,29)(26,79,32,73)(28,81,34,75)(30,83,36,77)(37,72,43,66)(39,62,45,68)(41,64,47,70)(50,63,56,69)(52,65,58,71)(54,67,60,61)(74,94,80,88)(76,96,82,90)(78,86,84,92), (1,76,67,18)(2,19,68,77)(3,78,69,20)(4,21,70,79)(5,80,71,22)(6,23,72,81)(7,82,61,24)(8,13,62,83)(9,84,63,14)(10,15,64,73)(11,74,65,16)(12,17,66,75)(25,46,86,50)(26,51,87,47)(27,48,88,52)(28,53,89,37)(29,38,90,54)(30,55,91,39)(31,40,92,56)(32,57,93,41)(33,42,94,58)(34,59,95,43)(35,44,96,60)(36,49,85,45), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,74,19,80)(14,79,20,73)(15,84,21,78)(16,77,22,83)(17,82,23,76)(18,75,24,81)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,54,43,60)(38,59,44,53)(39,52,45,58)(40,57,46,51)(41,50,47,56)(42,55,48,49)(61,66,67,72)(62,71,68,65)(63,64,69,70)(85,88,91,94)(86,93,92,87)(89,96,95,90)>;
G:=Group( (1,44,7,38)(2,55,8,49)(3,46,9,40)(4,57,10,51)(5,48,11,42)(6,59,12,53)(13,85,19,91)(14,31,20,25)(15,87,21,93)(16,33,22,27)(17,89,23,95)(18,35,24,29)(26,79,32,73)(28,81,34,75)(30,83,36,77)(37,72,43,66)(39,62,45,68)(41,64,47,70)(50,63,56,69)(52,65,58,71)(54,67,60,61)(74,94,80,88)(76,96,82,90)(78,86,84,92), (1,76,67,18)(2,19,68,77)(3,78,69,20)(4,21,70,79)(5,80,71,22)(6,23,72,81)(7,82,61,24)(8,13,62,83)(9,84,63,14)(10,15,64,73)(11,74,65,16)(12,17,66,75)(25,46,86,50)(26,51,87,47)(27,48,88,52)(28,53,89,37)(29,38,90,54)(30,55,91,39)(31,40,92,56)(32,57,93,41)(33,42,94,58)(34,59,95,43)(35,44,96,60)(36,49,85,45), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,74,19,80)(14,79,20,73)(15,84,21,78)(16,77,22,83)(17,82,23,76)(18,75,24,81)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,54,43,60)(38,59,44,53)(39,52,45,58)(40,57,46,51)(41,50,47,56)(42,55,48,49)(61,66,67,72)(62,71,68,65)(63,64,69,70)(85,88,91,94)(86,93,92,87)(89,96,95,90) );
G=PermutationGroup([(1,44,7,38),(2,55,8,49),(3,46,9,40),(4,57,10,51),(5,48,11,42),(6,59,12,53),(13,85,19,91),(14,31,20,25),(15,87,21,93),(16,33,22,27),(17,89,23,95),(18,35,24,29),(26,79,32,73),(28,81,34,75),(30,83,36,77),(37,72,43,66),(39,62,45,68),(41,64,47,70),(50,63,56,69),(52,65,58,71),(54,67,60,61),(74,94,80,88),(76,96,82,90),(78,86,84,92)], [(1,76,67,18),(2,19,68,77),(3,78,69,20),(4,21,70,79),(5,80,71,22),(6,23,72,81),(7,82,61,24),(8,13,62,83),(9,84,63,14),(10,15,64,73),(11,74,65,16),(12,17,66,75),(25,46,86,50),(26,51,87,47),(27,48,88,52),(28,53,89,37),(29,38,90,54),(30,55,91,39),(31,40,92,56),(32,57,93,41),(33,42,94,58),(34,59,95,43),(35,44,96,60),(36,49,85,45)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,74,19,80),(14,79,20,73),(15,84,21,78),(16,77,22,83),(17,82,23,76),(18,75,24,81),(25,32,31,26),(27,30,33,36),(28,35,34,29),(37,54,43,60),(38,59,44,53),(39,52,45,58),(40,57,46,51),(41,50,47,56),(42,55,48,49),(61,66,67,72),(62,71,68,65),(63,64,69,70),(85,88,91,94),(86,93,92,87),(89,96,95,90)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 5 | 3 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [5,5,0,0,0,0,3,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,0,0,0,0,2,1,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,5,8,0,0,0,0,5,0,0,0,0,0,0,0,8,5,0,0,0,0,8,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,5,8,0,0,0,0,0,0,8,0,0,0,0,0,8,5] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4Q | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | 2- (1+4) | Q8.15D6 | S3×C4○D4 | Q8○D12 |
| kernel | C42.154D6 | C4×Dic6 | C42⋊3S3 | Dic6⋊C4 | C12⋊Q8 | Dic3.Q8 | C12.3Q8 | C4⋊C4⋊7S3 | D6⋊Q8 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | Dic3 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 6 | 4 | 2 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{154}D_6 % in TeX
G:=Group("C4^2.154D6"); // GroupNames label
G:=SmallGroup(192,1255);
// by ID
G=gap.SmallGroup(192,1255);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,555,100,1571,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations